The stresses on an airship envelope, both on the supporting members and on the envelope fabric, come in two varieties: hoop stress and suspension stress.
Hoop stress is the stress on the walls of a container, typically having a generally cylindrical cross section, when the pressure inside the container exceeds the pressure outside. This stress tends to cause the container to burst. The name comes from the hoops found on wooden barrels that are used to hold in the staves and withstand this type of stress.
It is a well established fact of mechanics that the magnitude of hoop stress increases linearly with the diameter of the cylinder as well as linearly with the pressure difference between the inside and outside.
Suspension stresses are essentially the inverse of hoop stresses. These stresses arise when the pressure outside the container exceed the pressure inside, with the pressure tending to make the container collapse or implode. As with hoop stress, the magnitude of suspension stresses increases linearly with the pressure difference between the inside and outside. The relationship between the diameter of the container and the suspension stress is not as straightforward and not particularly relevant to the discussion here.
For a conventional pressurized airship at rest, all of the stresses on the envelope are the hoop stress. Further, these hoop stresses are greatest at the point of maximum diameter—typically at a point generally midway between the nose and tail of the ship.
At rest, the pressure distribution along the outside of an airship is at a uniform, ambient level. This pressure distribution changes as the airship starts to move through the air. However, the changes in external pressure are not straightforward. In fact, the pressure along the surface varies continuously along the direction of motion, typically parallel with the longitudinal axis of the envelope. At some points, pressure increases to a level greater than the ambient pressure. These are called “positive” aerodynamic forces. At other points, the pressure decreases to levels lower than the ambient pressure. These are called “negative” aerodynamic pressures. They are also sometimes referred to as “suction.”
As with all aerodynamic forces, the pressures created (both positive and negative) increase in magnitude as the square of the velocity of the airship. So, when the speed of the craft is doubled, the resulting stresses (both hoop and suspension) increase by a factor of 4.
At the forward-most location, typically the very nose of the ship (called the forward stagnation point,) the outside pressure increases well above the ambient pressure and thus produces more of an inward (positive) force. In a conventional pressurized envelope, this positive pressure actually reduces hoop stress on the nose portion of the envelope. In fact, at a high enough airspeed, the positive pressure will exceed the internal gas pressure making the nose of the envelope want to buckle inward. At such a high speed, the nose ceases to sustain hoop stress and starts to sustain suspension stress.
The positive (inward) pressure created by the airflow is at its greatest at the nose of the ship. As the air flows back along the outside of the envelope, the magnitude of the inward force rapidly decreases. In fact, by the time the airflow is typically about one tenth of the way back towards the tail (i.e. 10% of the way along the direction of motion, typically the longitudinal axis,) the relatively positive pressure completely disappears. This creates a zero-crossing point where the external pressure remains essentially unchanged at the initial, ambient atmospheric level.
As the airflow continues along the outside of the envelope, the pressure continues to decrease, and can reach a level below the ambient level and thus create a negative (outward or aerodynamic stress which increases the hoop stress on the envelope material.
At the widest part of the ship (typically about halfway between nose and tail) the magnitude of the change pressure is between one half and one third of that found the nose of the ship—but obviously in the opposite direction.
After the midpoint of the ship, the airflow start to re-converge and likewise the external pressure starts to return to ambient (the magnitude of the suction decreases.) Depending upon the exact shape of the tail of the ship and other factors, the air pressure may remain slightly on the side of suction, drop to essentially ambient pressure, or cross back over to a positive pressure at the tail. The stresses along the tail are much smaller in magnitude (and thus much easier to support structurally) than the stresses on the nose and around the middle of the ship.
It is most inconvenient that in a typical lighter-than-air airship reduction in ambient pressure (with respect to the internal pressure) created by the airflow is greatest around the middle of the ship, just where the hoop stress is already relatively highest due to the larger diameter of the ship at that point.
The problem of hoop stress is quite severe for designs that use the conventional method of deliberately increasing the internal pressure of the envelope (so-called pressure ships) in order to provide structural support for the envelope. Since the hoop stress is linearly related to the difference in pressure between the inside and the outside of the envelope, increasing the internal pressure must necessary increase the hoop stress by a comparable amount. By using a structurally reinforced envelope, hoop stresses may be reduced compared to a pressure ship design since the pressure inside the envelope doesn't need to be artificially increased above the ambient level in order to have the envelope retain its shape. Adding structural elements to stiffen the envelope also, unfortunately, adds weight to the airship.
What is needed is a way to minimize the stresses induced by pressure differentials between the internal and external sides of a flexibly covered vessel in motion while at the same time minimizing the weight of the vessel.